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COMPOSITE MAPPINGSHUSEN DING AND BING LIU Received 18 September 2005; Accepted 24 October 2005 We first obtain an improved version of the H¨older inequality with Orlicz norms.. Then, as

COMPOSITE MAPPING

SHUSEN DING AND BING LIU

Received 18 September 2005; Accepted 24 October 2005

We first obtain an improved version of the H¨older inequality with Orlicz norms Then,

as an application of the new version of the H¨older inequality, we study the integrability

of the Jacobian of a composite mapping Finally, we prove a norm comparison theorem

Copyright © 2006 S Ding and B Liu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Carl Gustav Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations into

a powerful tool for evaluating multiple integrals and solving differential equations Since then, the Jacobian (determinant) has played a critical role in multidimensional analysis and related fields, including nonlinear elasticity, weakly differentiable mappings, con-tinuum mechanics, nonlinear PDEs, and calculus of variations The integrability of Ja-cobians has become a rather important topic in the study of JaJa-cobians because one of the major applications of Jacobians is to evaluate multiple integrals Higher integrabil-ity properties of the Jacobian first showed up in [2], where Gehring invented reverse H¨older inequalities and used these inequalities to establish theL1+ε-integrability of the Jacobian of a quasiconformal mapping,ε > 0 Recently, the integrability of Jacobians of

orientation-preserving mappings of Sobolev classWloc1,n(Ω,Rn) has attracted the atten-tion of mathematicians, see [1,3–7], for instance The purpose of this paper is to study theL p(logL) α(Ω)-integrability of the Jacobian of a composite mapping

Let 0< p < ∞andα ≥0 be real numbers and letE be any subset ofRn We define the functional on a measurable function f over E by

[f ] L p(logL) α(E) =



E | f | plogα



e + | f |

 f  p



dx

 1/ p

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 89134, Pages 1 9

DOI 10.1155/JIA/2006/89134

where f  p =(

E | f (x) | p dx)1/ p In this paper, we always assume thatΩ is a bounded open subset ofRn,n ≥2 We writeL p(logL) α(Ω) for the space of all measurable functions

f on Ω such that [ f ]L p(logL) α(Ω)< ∞ As usual, we simply writeL p(Ω)= L1(logL)0(Ω) and

L log L(Ω)= L1(logL)1(Ω), respectively

A continuously increasing functionϕ : [0, ∞]→[0,∞] withϕ(0) =0 andϕ( ∞)= ∞is called an Orlicz function The Orlicz spaceL ϕ(Ω) consists of all measurable functions f

onΩ such that



Ωϕ  | f | λ



for someλ = λ( f ) > 0 L ϕ(Ω) is equipped with the nonlinear Luxemburg functional

 f  ϕ =inf



λ > 0 :



Ωϕ  | f | λ



dx ≤1

A convex Orlicz functionϕ is often called a Young function If ϕ is a Young function,

then ·  ϕ defines a norm inL ϕ(Ω), which is called the Luxemburg norm For ϕ(t) =

t plogα(e + t), 0 < p < ∞andα ≥0, we have

 f  L plogα L =inf



k :



Ω| f | plogα



e + | f | k



dx ≤ k p

FromTheorem 4.2 that will be proved later in this paper, we see that the Luxemburg norm f  ϕis equivalent to [f ] L p(logL) α( Ω) defined in (1.1) for any 0< p < ∞andα ≥0 Hence, the Orlicz spaceL ψ(Ω) with ψ(t)= t plogα(e + t) can be denoted by L p(logL) α(Ω) and the corresponding norm can also be written as [f ] L p(logL) α(Ω) The following version

of H¨older inequality appears in [3, Proposition 2.2]

Theorem 1.1 Let 1 < p, q < ∞ , α, β > 0, 1/ p + 1/q =1/r, α/ p + β/q = γ/r and f ∈

L p(logL) α(Ω), g∈ L q(logL) β(Ω) Then fg∈ L r(logL) γ(Ω) and

 f g  L rlogγ L ≤ C  f  L plogα L  g  L qlogβ L (1.5)

In this paper, we improve the condition 1< p, q < ∞into 0< p, q < ∞inTheorem 2.1

We enjoy the elementary method used in the proof ofTheorem 2.1 Then, using the im-proved H¨older inequality, we study theL p(logL) α(Ω)-integrability of the Jacobian of the composition of mappings

2 Improved H¨older inequality

UsingTheorem 1.1and the basic properties of logarithmic functions, we have the follow-ing generalized H¨older inequality

Theorem 2.1 Let m, n, α, β > 0, 1/s =1/m + 1/n, α/m + β/n = γ/s Assume that f ∈ L m

(logL) α(Ω) and g∈ L n(logL) β(Ω) Then, fg∈ L s(logL) γ(Ω) and



Ω| f g | slogγ



e + | f g |

 f g  s



dx

 1/s

≤ C



Ω| f | mlogα



e + | f |

 f  m



dx

 1/m

Ω| g | nlogβ



e + | g |

 g  n



dx

 1/n

, (2.1)

where C is a positive constant.

Note that (2.1) can be written as

[f g] L s(logL) γ(Ω)≤ C[ f ] L m(logL) α(Ω)[g] L n(logL) β(Ω). (2.2)

Proof Using the elementary inequality log(e + x a)≤log(e + x) a+1fora > 0, x > 0, we have

log



e + | f | s | g | s 1/s

| f | s | g | s 1/s

1



≤log



e + | f | s | g | s | f | s | g | s

1

 1/s+1

=

1

s + 1

 log



e + | f | s | g | s | f | s | g | s

1

 ,

log



e +



| f |

 f  m

s

≤log



e + | f |

 f  m

s+1

≤(s + 1) log



e + | f |

 f  m

 ,

log



e +



| g |

 g  n

s

≤log



e + | g |

 g  n

s+1

≤(s + 1) log



e + | g |

 g  n



.

(2.3)

From H¨older inequality (1.5) with 1=1/m/s + 1/n/s (note that m/s > 1, n/s > 1 since

1/s =1/m + 1/n) and (2.3), we have



Ω| f g | slogγ



e + | f g |

 f g  s



dx

=



Ω | f | s | g | s

logγ



e +  f | s | g | s |1/s | f | s | g | s 1/s

1



dx

≤ C1



Ω | f | s | g | s

logγ



e + | f | s | g | s | f | s | g | s

1



dx

≤ C2



Ω | f | sm/s

logα



e+ | f | s | f | s m/s



dx

s/m 

Ω | g | sn/s

logβ



e+ | g | s | g | s n/s



dx

s/n

= C2



Ω| f | mlogα



e+



| f |

 f  m

s

dx

s/m 

Ω| g | nlogβ



e+



| g |

 g  n

s 

dx

s/n

≤ C3



Ω| f | mlogα



e + | f |

 f  m



dx

s/m 

Ω| g | nlogβ



e + | g |

 g  n



dx

s/n

.

(2.4)

Hence, we conclude that



Ω| f g | slogγ



e + | f g |

 f g  s



dx

 1/s

≤ C4



Ω| f | mlogα



e + | f |

 f  m



dx

 1/m

Ω| g | nlogβ



e + | g |

 g  n



dx

 1/n

(2.5)

FromTheorem 2.1, we have the following general result immediately

Corrollary 2.2 Let p i > 0, α i > 0 for i =1, 2, , k, 1/ p1+ 1/ p2+···+1/ p k =1/ p, and

α1/ p1+α2/ p2+···+α k / p k = α/ p Assume that f i ∈ L p i(logL) α i(Ω) for i=1, 2, , k Then

f1f2··· f k ∈ L p(logL) α(Ω) and

f1f2··· f k

L p(logL) α( Ω)≤ C f1



L p1(logL) α1( Ω) f2



L p2(logL) α2( Ω)··· f k

L pk(logL) αk(Ω), (2.6)

where C is a positive constant and the norms [ f1f2··· f k]L p(logL) α(Ω) and [ f i]L pi(logL) αi(Ω),

i =1, 2, , k, are defined in (1.1).

3 Integrability of Jacobians of composite mappings

In this section, we explore applications of the new version of the H¨older inequality estab-lished in the last section Specifically, we study the integrability of the Jacobian of the com-position of mappings f :Ω→ R n, f =(f1(u1,u2, , u n), f2(u1,u2, , u n), , f n(u1,u2,

, u n)) of Sobolev class Wloc1,p(Ω,Rn), whereu i = u i( x1,x2, , x n), i =1, 2, , n, are

func-tions ofx =(x1,x2, , x n)∈ Ω with continuous partial derivatives ∂ui /∂x j, =1, 2, , n.

Assume that the distributional differential D f (u)=[∂ f i /∂u j] and Du(x) =[∂u i /∂x j] are

locally integrable functions with values in the space GL(n) of all n × n-matrices As usual,

we write

J(x, f ) =detD f (u(x)) = ∂ f1··· f n

J(u, f ) =detD f (u) = ∂ f1··· f n

J(x, u) =detDu(x) = ∂ u1··· u n

∂ x1··· x n

respectively UsingTheorem 2.1, we have the following integrability theorem for the Ja-cobian of the composition of mappings

Theorem 3.1 Let s, t, β, γ > 0, with 1/ p =1/s + 1/t and β/s + γ/t = α/ p Assume that J(x, f ), J(u, f ), and J(x, u) are Jacobians defined in (3.1), (3.2), and (3.3), respectively If

J(u(x), f ) ∈ L s(logL) β(Ω) and J(x,u)∈ L t(logL) γ(Ω), then J(x, f )∈ L p(logL) α(Ω) and



Ω

J(x, f )p

logα



e + J(x, f )

J(x, f )

p



dx

 1/ p

≤ C



Ω

J(u, f )s

logβ



e + J(u, f )

J(u, f )

s



dx

 1/s

×



Ω

J(x, u)t

logγ



e + J(x, u)

J(x, u)

t



dx

 1/t

,

(3.4)

where C is a positive constant.

Proof Note that the Jacobian of the composition of f and u can be expressed as

J(x, f ) = ∂ f1··· f n

∂ x1··· x n = ∂( f1··· f n)

∂ u1··· u n · ∂ u1··· u n

∂ x1··· x n = J(u, f ) · J(x, u). (3.5) ApplyingTheorem 2.1and (3.5) yields



Ω

J(x, f )p

logα



e + J(x, f )

J(x, f )

p



dx

 1/ p

=



Ω

J(u, f ) · J(x, u)p

logα



e + J(u, f ) · J(x, u)

J(u, f ) · J(x, u)

p



dx

 1/ p

≤ C



Ω

J(u, f )s

logβ



e + J(u, f )

J(u, f )

s



dx

 1/s

×



Ω

J(x, u)t

logγ



e + J(x, u)

J(x, u)

t



dx

 1/t

< ∞

(3.6)

sinceJ(u(x), f ) ∈ L s(logL) β(Ω) and J(x,u)∈ L t(logL) γ(Ω) Thus, J(x, f )∈ L p(logL) α(Ω)

Applying the H¨older inequality withL p-norms

 f g  s,E ≤  f  α,E ·  g  β,E, (3.7)

where 0< α, β < ∞,s −1= α −1+β −1, and f and g are any measurable functions on a

measurable setE ⊂ R n, we have the followingL p-integrability theorem for the Jacobian

of a composite mapping

Theorem 3.2 Let J(x, f ), J(u, f ), and J(x, u) be the Jacobians defined in (3.1), (3.2), and (3.3), respectively If J(u(x), f ) ∈ L s(Ω) and J(x,u) ∈ L t(Ω), s,t > 0, then J(x, f ) ∈ L p(Ω)

and

 J(x, f )  L p(Ω)≤ C J(u(x), f )

L s(Ω) J(x, u)

where C is a positive constant and the integrability exponent p of J(x, f ) determined by

1/ p =1/s + 1/t is the best possible.

The following example shows that the integrability exponent p of J(x, f ) cannot be

improved anymore

Example 3.3 We consider the mappings

f (x, y) = f1, 2

=



x

x2+y2 σ, y

x2+y2 σ

 , (x, y) ∈ D =(x, y) : 0 < x2+y2≤ ρ2

,

x = r − kcosθ, y = r − ksinθ, (r, θ) ∈Ω= {(r, θ) : 0 < r < ρ, 0 < θ ≤2π },

(3.9)

whereσ and ρ are positive constants After a simple calculation, we obtain the following

Jacobians:

J1= ∂( f1, 2)

∂(r, θ) = k(2σ −1)

r4σ+2k+1 , J2= ∂( f1, 2)

∂(x, y) =1−2

r4σ ,

J3= ∂(x, y) ∂(r, θ) = r −2k+1 k , 0< r < ρ.

(3.10)

It is easy to see thatJ1∈ L1/(4σ+2k+1)(Ω) but J1 ∈ L p(Ω) for any p > 1/(4σ + 2k + 1)

Sim-ilarly,J2∈ L1/4σ(Ω) but J2 ∈ L s(Ω) for any s > 1/4σ and J3∈ L1/(2k+1)(Ω) but J3 ∈ L t(Ω) for anyt > 1/(2k + 1) Here, the integrability exponent p =1/(4σ + 2k + 1) of ∂( f1, 2)/∂

(r, θ) is determined by

1

p =(4σ + 2k + 1) =1

s +

1

where s =1/4σ and t =1/(2k + 1) are the integrability exponents of Jacobians

∂( f1, 2)/∂(x, y) and ∂(x, y)/∂(r, θ), respectively.

The above example shows that, inTheorem 3.2, the integrability exponentp of J(x, f )

that is determined by 1/ p =1/s + 1/t is the best possible, where s is the integrability

expo-nent ofJ(u(x), f ) and t is the integrability exponent of J(x, u).

Example 3.4 Let J1= ∂( f1, 2)/∂(r, θ), J2= ∂( f1, 2)/∂(x, y), and J3= ∂(x, y)/∂(r, θ) be

the Jacobians obtained inExample 3.3 For anyε > 0, there exists a constant C1> 0 such

that

J1p

log



e + J1

J1

p



≤ C1 J1p+ε/(4σ+2k+1)

Using (3.10) and (3.12), we have



Ω

J1p

log



e + J1

J1

p



drdθ

=2π

ρ

0

J1p

log



e + J1

J1

p



dr

= C2

ρ

0

r4σ+2k+1

p log



e + (2σ −1)k/r4σ+2k+1

(2σ −1)k/r4σ+2k+1

p



dr

≤ C3

ρ

0r −(4σ+2k+1)p r −(4σ+2k+1)ε/4σ+2k+1

dr ≤ C4

ρ

0 r −(4σ+2k+1)p − ε dr = C5< ∞

(3.13)

for anyp satisfying 0 < p ≤1/(4σ + 2k + 1) − ε/(4σ + 2k + 1) Since ε > 0 is arbitrary, we

know thatJ1∈ L plogL( Ω) for any p with 0 < p < 1/(4σ + 2k + 1) Similarly, we have J2∈

L slogL( Ω) for any s with 0 < s < 1/4σ and J3∈ L tlogL( Ω) for any t with 0 < t < 1/(2k + 1).

This example shows that the integrability exponent p of ∂( f1, 2)/∂(r, θ) that is

deter-mined by 1/ p =1/s + 1/t is the best possible when α = β = γ =1 inTheorem 3.1

4 The norm comparison theorem

In this section, we discuss the relationship between norms f  L plogα Land [f ] L p(logL) α( Ω), which will provide a different way to prove Theorems 2.1and 3.1 First, we recall the following more general inequality appearing in [3, Theorem A.1]

Theorem 4.1 Suppose that A, B, C : [0, ∞)→[0,∞ ) are continuous, monotone increasing functions for which there exist positive constants c and d such that

(i)B −1(t)C −1(t) ≤ cA −1(t) for all t > 0,

(ii)A(t/d) ≤1/2A(t) for all t > 0.

Suppose that G is an open subset ofRn , for f ∈ L B( G) and g ∈ L C( G) Then f g ∈ L A( G) and

In [6], Iwaniec and Verde prove that the norm f  L plogα L is equivalent to the norm [f ] L p(logL) α(Ω)for 1< p < ∞ Similar to the proof of [6, Lemma 8.6], we have the relation-ship between the norm f  L plogα Land the norm [f ] L p(logL) α( Ω)

Theorem 4.2 For each f ∈ L p(logL) α(Ω), 0 < p <∞ and α ≥ 0,

 f  p ≤  f  L plogα L ≤[f ] L p(logL) α( Ω)≤ C  f  L plogα L, (4.2)

where C =2α/ p(1 + (α/ep) α)1/ p is a constant independent of f

Proof Let K =  f  L plogα L Then, by the definition of the Luxemburg norm, we have

K =



Ω| f | plogα



e + | f | K



dx

 1/ p

It is clear thatK ≥  f  pand

K ≤



Ω| f | plogα



e + | f |

 f  p



dx

 1/ p

=[f ] L p(logL) α(Ω), (4.4)

that is,

 f  L plogα L ≤[f ] L p(logL) α( Ω). (4.5)

On the other hand, usingK ≥  f  p and the elementary inequality| a + b | s ≤2s(| a | s+

| b | s),s ≥0, we obtain that



Ω| f | plogα



e + | f |

 f  p



dx =



Ω| f | plogα



e + | f |

 f  p



dx

≤



Ω| f | p

 log



e + | f | K

 + log



K

 f  p

α

dx

≤2α



Ω| f | plogα



e + | f | K

 + 2α



Ω| f | plogα



K

 f  p



dx

=2α K p+ 2α  f  p plogα



K

 f  p



.

(4.6) Note that the functionh(t) = t plogα(K/t), 0 < t ≤ K, has its maximum value (α/ep) α K p

att = K/e α/ p Then

 f  p plogα



K

 f  p



≤



α ep

α

Combining (4.6) and (4.7) gives



Ω| f | plogα



e + | f |

 f  p



dx ≤2α



1 +

 α

ep

α

which is equivalent to

[f ] L p(logL) α(Ω)≤ C  f  L plogα L, (4.9) whereC =2α/ p(1 + (α/ep) α)1/ p The proof ofTheorem 4.2has been completed 

It is easy to see thatTheorem 4.2indicates that, for any 0< p < ∞andα ≥0, the Lux-emburg norm f  L plogα Lis equivalent to the norm [f ] L p(logL) α(Ω)defined in (1.1) Hence,

we can also prove Theorems2.1and3.1usingTheorem 4.1with suitable choices of func-tionsA(t), B(t), and C(t).

[1] H Brezis, N Fusco, and C Sbordone, Integrability for the Jacobian of orientation preserving

map-pings, Journal of Functional Analysis 115 (1993), no 2, 425–431.

[2] F W Gehring, The L p -integrability of the partial derivatives of a quasiconformal mapping, Acta

Mathematica 130 (1973), 265–277.

[3] J Hogan, C Li, A McIntosh, and K Zhang, Global higher integrability of Jacobians on bounded

domains, Annales de l’Institut Henri Poincar´e Analyse Non Lin´eaire 17 (2000), no 2, 193–217.

[4] T Iwaniec and C Sbordone, On the integrability of the Jacobian under minimal hypotheses,

Archive for Rational Mechanics and Analysis 119 (1992), no 2, 129–143.

[5] , Weak minima of variational integrals, Journal f¨ur die reine und angewandte Mathematik

454 (1994), 143–161.

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(1999), no 2, 391–420.

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und angewandte Mathematik 412 (1990), 20–34.

Shusen Ding: Department of Mathematics, Seattle University, Seattle, WA 98122, USA

E-mail address:sding@seattleu.edu

Bing Liu: Department of Mathematical Sciences, Saginaw Valley State University, University Center,

MI 48710, USA

E-mail address:bliu@svsu.edu

3 Integrability of Jacobians of composite mappings

In this section, we explore applications of the new version of the Hăolder inequality estab-lished in the last section Specifically,... we study the integrability of the Jacobian of the com-position of mappings f :Ω→ R n, f =(f1(u1,u2,...

Assume that the distributional differential D f (u)=[∂ f i /∂u j] and Du(x) =[∂u i /∂x j] are